Abstract
A bounded everywhere defined operator T in a Hilbert space ℌ is said to be a quasi-selfadjoint contraction or (for short) a qsc-operator, if T is a contraction and ker (T − T*) ≠ {0}. For a closed linear subspace \( \mathfrak{N} \) of ℌ containing ran (T − T*) the operator-valued function QT (z)=\( P_\mathfrak{A} \) (T − zI)−1↾\( \mathfrak{N} \) , |z|>1, where P\( P_\mathfrak{A} \) is the orthogonal projector from ℌ onto \( \mathfrak{N} \) , is said to be a Q-function of T acting on the subspace \( \mathfrak{N} \) . The main properties of such Q-functions are studied, in particular the underlying operator-theoretical aspects are considered by using some block representations of the contraction T and analytical characterizations for such functions QT(z) are established. Also a reproducing kernel space model for QT(z) is constructed. In the special case where T is selfadjoint QT(z) coincides with the Q-function of the symmetric operator A ≔ T↾ (ℌ ⊖ \( \mathfrak{N} \) ) and its selfadjoint extension T = T* in the usual sense.
To Heinz Langer on the occasion of his retirement
This work was supported by the Research Institute for Technology at the University of Vaasa. The first author was also supported by the Academy of Finland (projects 203227, 208057) and the Dutch Organization for Scientific Research NWO (B 61-553).
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Arlinskii, Y., Hassi, S., de Snoo, H. (2005). Q-functions of Quasi-selfadjoint Contractions. In: Langer, M., Luger, A., Woracek, H. (eds) Operator Theory and Indefinite Inner Product Spaces. Operator Theory: Advances and Applications, vol 163. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7516-7_2
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